With the aims of understanding the mechanisms of maintenance of genetic variablity and evolution of populations and utilizing the genetic polymorphism in man for medical and forensic purposes, three coherent research projects are proposed. (1) Genetic differentiation of populations. Mathematical theories of the evolutionary change of restriction site pattern of DNA, probability distribution of genetic distance, and evolutionary process of the establishment of reproductive isolation in speciation will be developed. Statistical methods of constructing phylogenetic trees or denrograms by using gene frequency data will also be studied. In addition to these theoretical studies data on the interspecific or interracial differences of restriction site maps in mitrochondrial and nuclear DNAs will be analyzed by using the statistical methods newly developed. All available data on protein and DNA polymorphisms will be compiled from many different human populations, and the pattern and mechanism of evolution of human races will be investigated. (2) Genetic variability and its maintenance in populations. Mathematical models for studying DNA polymorphisms in terms of restriction enzymes will be studied by considering synonymous and nonsynonymous substitutions, the difference in the rate of nucleotide substitutions among the three nucleotide positions of codons, and deletion and insertion. Mathematical studies will also be conducted about the effects of selection, particularly fluctuating selection, on the amount and pattern of protein polymorphism in natural populations. (3) Utility of marker genes for genetic counseling and paternity test. Mathematical methods will be developed for computing the risk of genetic diseases and the proportion of informative families when restriction site markers in DNA are used for genetic counseling. Mathematical methods for computing the probabilities of excluding and identifying paternity for a given pair of mother and child by using genetic markers will also be studied.